Identification of hidden Markov chains governing dependent credit-rating migrations

Three models of dependent credit-rating migrations are considered. Each of them entails a coupling scheme and a discrete-time Markovian macroeconomic dynamics. Every credit-rating migration is modeled as a mixture of an idiosyncratic and a common component. The larger is the pool of debtors affected by the same common component, the stronger is the dependence among migrations. The distribution of the common component depends on macroeconomic conditions. At every time instant, the resulting allocation of debtors to credit classes and industries follows a mixture of multinomial distributions. Dealing with M non default credit classes, there are 2M theoretically possible macroeconomic outcomes. Only few of them occur with a positive probability. Restricting the macroeconomic dynamics to such outcomes simplifies estimation. A heuristics for identifying them is suggested. Using the maximum likelihood method, it was tested on a Standard and Poor's (S&P's) data set.